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Properties

Label 2-1800-72.43-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.173 - 0.984i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + 2·17-s + (−0.5 + 0.866i)18-s − 19-s + (−0.999 − 1.73i)22-s + 0.999·24-s + 27-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + 2·17-s + (−0.5 + 0.866i)18-s − 19-s + (−0.999 − 1.73i)22-s + 0.999·24-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192115183\)
\(L(\frac12)\) \(\approx\) \(1.192115183\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 2T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.636244243761184077375394631291, −8.703877462254090330272501374774, −7.937530005884508882877818960151, −7.48973060725836283519371668778, −6.81820695663049927129490477170, −5.63962102710827890834517924831, −4.80485668233113411519614739567, −3.97339274868367164000492665254, −2.60272569228288430237098657724, −1.56022636348637331822896310272, 1.07402387517953136337520212575, 2.43735013923784768265377853053, 3.19115942582971318824295854572, 3.81849036355347273009978678623, 5.02773404103110814865024471869, 6.06791180138060135031911495590, 7.42640728268411725935772014362, 8.048399953227304205174598572948, 8.461331675971237753038636605050, 9.301220454314294819473693637163

Graph of the $Z$-function along the critical line